Theoretical and computational physical chemistry group

Theoretical characterization of Theoretical characterization of -helix -helix

and and -hairpin folding kinetics-hairpin folding kinetics

Isabella Daidone

Cambridge, 26/07/2005

Theoretical and computational physical chemistry group

Free energy

ProteinProtein foldingfolding

Theoretical and computational physical chemistry group

Folding kineticsFolding kinetics

[ms - minutes] [s timescale]

Proteins

Peptides

[10 ns – hundreds ns] [hundreds ns - s]

C-terminal -hairpin of GB1 protein 6 speptide I 760 ns

alanine-rich peptides

Theoretical and computational physical chemistry group

Folding kineticsFolding kinetics

folding kinetics is still computationally prohibitive !!!

Peptides

[10 ns – hundreds ns][hundreds ns - s]

Molecular Dynamics (MD) simulations

[hundreds of ns]

•Simplified protein models

•Massive parallel computing

•Transition path sampling

•Diffusive rate theory

Theoretical and computational physical chemistry group

( ) ( , ) ( )k kP t q t q dqr c= ò

( , ) ( , ) ( ) ( , )q t q t A q q tD

t q q q RT

(q,t), peptide probability density

Pk(t), time probability of secondary structure states, k

equilibrium fraction of secondary structure states, k

( )

7( )

0

( )k

k

A q

kA q

k

eq

e

b

bc

- D

- D

=

=

å

( )( ) ln

( )eq

eq ref

qA q RT

qr

rD = - external potential given

by the free energy

A (

q)

folding reaction coordinate, q

Theoretical and computational physical chemistry group

MD simulation

• Starting structure• Equations of motion (force field)

Kinetic simulation

• Starting condition of the system• Diffusion equations (free energy gradient)

A (

kJ/

mol)

q

Theoretical and computational physical chemistry group

Folding kinetic simulationFolding kinetic simulation

• Proper reaction coordinates, q

• Free energy profile, A(q) , and k(q)

• Diffusion coefficient, D

Theoretical and computational physical chemistry group

Folding kinetic simulationFolding kinetic simulation

• Proper reaction coordinates, q

• Free energy profile, A(q) , and k(q)

• Diffusion coefficient, D

Theoretical and computational physical chemistry group

““Reaction coordinates”: essential degrees of Reaction coordinates”: essential degrees of freedomfreedom

A. Amadei et al., PROTEINS, 17:412-425, 1993. Essential dynamics of proteins

Theoretical and computational physical chemistry group

““Reaction coordinates”: essential degrees of Reaction coordinates”: essential degrees of freedomfreedom

A. Amadei et al., PROTEINS, 17:412-425, 1993. Essential dynamics of proteins

ij i i j j C x x x x Positional fluctuationsPositional fluctuationscovariance matrixcovariance matrix

Eigenvectors of fluctuationsEigenvectors of fluctuationsand corresponding eigenvaluesand corresponding eigenvaluesCTTΛ T

q first essential eigenvector

Theoretical and computational physical chemistry group

Folding kinetic simulationFolding kinetic simulation

• Proper reaction coordinates, q

• Free energy profile, A(q) , and k(q)

• Diffusion coefficient, D

Theoretical and computational physical chemistry group

Thermodynamic propertiesThermodynamic properties

( ) ( )( ) ln ln

( ) ( )eq

eq ref ref

q N qA q RT RT

q N qr

rD = - -;

( )

7( )

0

( )( )

( )

k

k

A qk

kA q

k

N qeq

N qe

b

bc

- D

- D

=

=

å;

A (

q)

q

Theoretical and computational physical chemistry group

Folding kinetic simulationFolding kinetic simulation

• Proper reaction coordinates, q

• Free energy profile, A(q) , and k(q)

• Diffusion coefficient, D

Theoretical and computational physical chemistry group

Free diffusion over the essential Free diffusion over the essential eigenvectoreigenvector

time (ns)

q (

nm

)

-3-3

-2-2

-1-1

00

11

Theoretical and computational physical chemistry group

D and D0 are the long and short-time diffusion constants, respectively

1 , 2, 3 are the relaxation times of the two switching modes

mean square displacement <q2(t)> (nm2/ps)

<q2(t)> = 2 t + 2 (D0-A1) 1 (1-exp(-t/1)) + 2 (D0-A2) 2 (1-exp(-t/2))

+ 2 (D0-A3) 3 (1-exp(-t/3))

D

Free diffusion over the essential Free diffusion over the essential eigenvectoreigenvector

slope=2D0

slope=2D00

Theoretical and computational physical chemistry group

Model systemsModel systems

Prion Protein H1 peptide (14 residues)

Antimicrobial temporin L (13 residues)

MKHMAGAAAAGAVV FVQWFSKFLGRIL

Amyloidogenic(-sheet rich firbrils)

-helical(circular dichroism)

Theoretical and computational physical chemistry group

MD simulationsMD simulations* of the H1 of the H1 peptidepeptide

•PME•N,V,T (isokinetic)

•periodic truncated octahedron•1 nm explicit solvent on all sides

*GROMACS software package

Total simulation time of 1.1 s

850300 water (SPC)-hairpin

240300water (SPC)-helix

Length (ns)Temp (K)SolventStarting structure

Theoretical and computational physical chemistry group

MD simulationsMD simulations* of temporin Lof temporin L

300300water (SPC)-helix

Length (ns)Temp (K)SolventStarting structure

•PME•N,V,T (isokinetic)

•periodic truncated octahedron•1 nm explicit solvent on all sides

*GROMACS software package

Theoretical and computational physical chemistry group

1

H1 peptideH1 peptide

+ + + + + + = 30% = 70%

Theoretical and computational physical chemistry group

5

temporin Ltemporin L

280240

Time (ns)

= 16%= 14%= 70%

Theoretical and computational physical chemistry group

A (

kJ/

mol)

q (nm) q (nm)

0 1 2 3 4 5 76 L=0 L=1 L=2

k q

L=1

L=0

L=2

Theoretical and computational physical chemistry group

Diffusion coefficientsDiffusion coefficients

2.4 10-5

(0.5 10-5)

-hairpin

3.2 10-5

(0.5 10-5)

-helix

Dnm2ps-1

system

<1

<1

1ps

0.026(0.001)

0.026(0.001)

D0nm2ps-1

7.8(1)

4.3(1)

2ps

113.4(2)

30.6(2)

3ps

Theoretical and computational physical chemistry group

Eigenvector I (nm)

(q,t)0

50

150

100

Tim

e (n

s)

0

1

-1-2

200

( ) ( , ) ( )k kP t q t q dqr c= òA

(kJ/

mol)

q (nm)

( , ) ( , ) ( ) ( , )q t q t A q q tD

t q q q RT

Theoretical and computational physical chemistry group

-hairpin mean folding time-hairpin mean folding time

fold 30 ns-helix mean folding time-helix mean folding time

fold 8 ns

Pk(q,t)Pk(q,t)

Time (ns)

+ + + + + +

Theoretical and computational physical chemistry group

s

V. Munoz et al., Nature, Folding dynamics and mechanism of -hairpin formation. 390:196-199, 1997.

TerminalTerminal-hairpin of Protein G-hairpin of Protein G

exp s

A# kJ/molA

(kJ/

mol)

q (nm)

Theoretical and computational physical chemistry group

q (nm)

A (

kJ/

mol)

-hairpin folding kinetics-hairpin folding kinetics

bi

one

tt

= 1.22

Daidone et al., 2005. Theoretical characterization of -helix and -hairpin folding kinetics, Manuscript in

preparation.

q1 (nm)q

2 (

nm

)

Theoretical and computational physical chemistry group

AcknowledgmentsAcknowledgments

Dr. Marco D’Abramo

Prof. Jeremy Smith (University of Heidelberg)

Prof. Alfredo Di Nola(University “La Sapienza” of Rome)

Dr. Andrea Amadei (University of Rome “Tor Vergata” )